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IEEE SIGNAL PROCESSING LETTERS, VOL. 11, NO. 9, SEPTEMBER 2004
Reduced Complexity Equalization Schemes for
Zero Padded OFDM Systems
M. R. Bhavani Shankar and K. V. S. Hari, Senior Member, IEEE
Abstract—In this work, three reduced complexity equalization
schemes for Zero-padded OFDM systems are described. These
schemes guarantee Zero-Forcing (ZF) equalization irrespective
of the channel nulls. Two of these schemes implement the minimum-norm ZF equalizer efficiently using QR decomposition. In
the third scheme, the channel zeros are grouped as being inside
or outside or on the unit circle. These groups are then equalized
sequentially in a manner so as to tackle excess noise amplification.
The three schemes are compared for their computational complexity and Bit Error Rate (BER) performance. It is shown that
the attractive scheme depends on the system specifications. The
BER- Computations trade off occurring in the choice of the right
algorithm is also highlighted.
Index Terms—Minimum–maximum phase, Orthogonal Frequency Division Multiplexing (OFDM), pseudo-inverse, Toeplitz
matrices, zero padding.
I. INTRODUCTION
T
HE use of Zero Padding (ZP) [1] in Orthogonal Frequency
Division Multiplexing (OFDM) systems provides for
equalizers that guarantee Zero Forcing Equalization (ZFEQN)
for all nonzero channels. The minimum-norm Zero Forcing
Equalizer (ZFE) given in (9) of [1] is one such equalizer.
Henceforth, the minimum-norm equalizer is referred to as the
Standard ZFE (SZFE). However, direct implementation of the
pseudo inverse solution is computationally expensive [3]. Fast
equalizers for ZP scheme are proposed in [2], which, unfortunately, do not guarantee ZFEQN for all channels. In this work
we describe three low complexity schemes that yield ZFEQN
for all nonzero channels. We begin with the use of QR Decomposition (QRD) [3] for efficient implementation of SZFE.
Since SZFE solves a Least Squares (LS) problem involving a
Toeplitz matrix, any of the fast algorithms in [4] can be used to
implement it. Since all the algorithms in [4] have similar orders
of complexity, we consider one of them—the BBH algorithm
[6], [4]—for implementing SZFE. A recent work by Sinn et
al. [5], uses an algorithm from [4] for obtaining an efficient
ZF algorithm in a slightly different context. We also show that
SZFE implemented using Householder’s method [3] can be
cheaper than the BBH algorithm under certain conditions. We
then propose the Min-Max equalization, which is based on
grouping the channel zeros as being inside, outside or on the
Manuscript received December 6, 2003; revised January 15, 2004. The associate editor coordinating the review of this manuscript and approving it for
publication was Dr. Geert Leus.
The authors are with the Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore, India (e-mail: bshankar@protocol.ece.iisc.ernet.in; hari@ece.iisc.ernet.in).
Digital Object Identifier 10.1109/LSP.2004.833491
unit circle and equalizing for these groups sequentially. These
equalizers are evaluated for their computational complexity and
Bit Error Rate (BER) performance. The choice of the attractive
algorithm and the BER performance-complexity trade-off
occurring therein are discussed.
II. EQUALIZATION USING ZERO PADDING
We consider a system model similar to the one presented in
be the base[1]. Let be the number of sub-carriers and
band channel transfer function with degree and co-efficients
. Let
and
be the data transIFFT) at th instant.
mitted on sub-carriers (output of
be the -fold blocked received vector at th instant
Let
and
be the
Additive White Gaussian Noise vector
at the receiver front-end. The synchronized receiver is assumed
to know the channel perfectly. The transmitter normally pads
zeros to
. For simplicity, we assume that
.
Let be the
Toeplitz matrix with
and
being its first row and column respectively [1]. The input-output relation and the SZFE [1]1 for
such a zero padded OFDM system are given by (1) and (2), respectively, as
(1)
(2)
The computations required for direct implementation of
is
[3]. Since FFT/IFFT operations are unitary, it suffices
to have an estimate of
. Further, (1) and hence the proposed
techniques are also valid for the general class of zero padded
block transmission systems [1].
III. IMPLEMENTING STANDARD ZFE USING QR
DECOMPOSITION
denote the QRD of [3]. Since any nonzero
Let
is full rank [1], the output of SZFE (
, (2)) is obtained as
the solution of
[3]. In this section, we employ
two known algorithms to find the QRD of efficiently, so that
can be obtained as above. Complexities of the implementations are also evaluated. For brevity, operations are abbreviated
as CA, CM, CD, RA, RM, RD, RCM, and RCD. Here, A stands
for additions (and subtractions), M for multiplications and D
for divisions. C (R) denotes that both the operands are complex
(real). RCx involves a real and a complex operand. Real complex operations are mentioned as they are cheaper than their
complex counterparts. Further, for a vector , we implement
1Operations:
1070-9908/04$20.00 © 2004 IEEE
denotes Transpose,
Hermitian and y Pseudo inverse.
IEEE SIGNAL PROCESSING LETTERS, VOL. 11, NO. 9, SEPTEMBER 2004
as
. Operations involving a zero at known
) are not counted.
positions (independent of
A. Bojanczyk, Brent, and de Hoog (BBH) Algorithm
The BBH algorithm [6] can be simplified to exploit the
sparseness of . A part of the simplified algorithm is described
in [5]. The simplified algorithm, in its entirety, is described
denote the
below for completeness. Let
sub-matrix formed by retaining rows through and columns
through of a matrix . Let
be the
th element of
and
be the th element of vector .
Initialization:
square-root, a RD,
RM,
for
. Initializations require a
CM,
CA,
RCM.
RA and
to
Step 1:
. (Note
.).
Step (2a):
.
Step (2b):
.
end
loop.
Step 3:
. This requires
CM and
CA. The presence of
nonzero entries in the th column of is exploited.
only
by Back Substitution. This
Step 4: Solve
CM,
step requires about
CA and CD when the low bandwidth
[3] of and positivity of
are exploited.
requires
RM,
RA,
RD,
Step 1
RCM and
square roots. Since
are taken to be
positive, ’s are real resulting in RCM. Step (2a)
requires
CM,
around
RCM,
RD, and
CA. Further, Step (2b)
requires about
CM,
RCM and
CA (computing even for
). The total number of
floating point operations (flops) [3] required by BBH method is
.
approximately
B. Householder’s Method
Apart from the fast algorithms, Householder’s, Givens and
other classical methods [3] can also be used to obtain QRD of
. We use the Householder’s method as a representative algorithm. Further, Householder’s method is shown to perform
QRD efficiently for some cases [3]. The Householder’s algorithm for finding full rank LS solution in [3] is used here with
simple modifications to account for the low bandwidth of .
Hence, we do not describe the algorithm here. Further, the implementation is as explained in [3]. Obtaining the parameters
requires approximately
of Householder’s matrix
RM,
RA,
Square roots,
RCM and
CD. Obtaining and
requires about
753
CM,
CA and
RCM. RCM arise because
[3] used is real. Finally, back substitution requires
CD,
CM and
CA because of the low bandwidth of . So this method approxflops. The fact that this alimately requires
gorithm uses sparseness effectively (for some cases), better than
BBH, to reduce computations is shown in Section V.
IV. MIN-MAX EQUALIZATION
To motivate this scheme, assume (1) noise-free case, (2)
. Let
channel has no zeros on the unit circle and (3)
be the first entries of
and
be the leading
Lower Triangular Toeplitz (LTT) sub-matrix of . Then
can be easily obtained by solving
, through
back substitution. Since
is also LTT [9],
is
by the entries in the
equivalent to filtering unblocked
(say
) and retaining the first
first column of
outputs.
can be shown to be the first
impulse
. In the presence of noise,
response co-efficients of
due to the implicit filtering involved, such an equalization is
(i.e., stable
).
viable only for minimum phase
were to be maximum phase, the above approach results
If
in undue noise amplification. Use of time reversed filtering
to
instead of the normal filtering, effectively converts
a minimum phase function, thereby preventing excess noise
amplification. The details of time reversed filtering and the
approach for a mixed phase system are mentioned below.
Consider a channel with no roots on the unit circle in
a noiseless setting. The case of zeros on the unit circle is
dealt later. In the presence of noise, Min-max equalization
scheme does not implement the SZFE. The performance
of this scheme vis-a-vis that of SZFE is discussed in Secand
tion V. Let
denote the co-efficients (transfer functions) of maximum
and minimum phase parts of the channel respectively. Then
. Let
be the
-fold blocked input with a zero pad of length (
zeros) for block transmissions over
and let
be the corresponding -fold blocked output. Then
is
the input for block transmission over
and let
be the corresponding -fold blocked output. Since
and
have and
trailing zeros respectively, output at
be the
LTT matrix
each stage is free from IBI. Let
with
being its first column.
is the
LTT matrix with
being its first column. Then,
(3)
It can be shown that
Hence we will use
equals
instead of
of (1) (noiseless case).
.
A. Equalizer for
Let
matrix
and
and
the
denote the
zero matrix
identity
respectively.
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IEEE SIGNAL PROCESSING LETTERS, VOL. 11, NO. 9, SEPTEMBER 2004
Let
and
be the
leading LTT sub-matrix of
.
Then,
(4)
Since
(minimum phase property),
is invertible.
Hence, (4) can be easily solved for
by back substituhas
trailing zeros, recovering
tion. As
performs equalization for the minimum phase part.
B. Equalizer for
Let
the
and
be
UTT (Upper Triangular Toeplitz) matrix with
being its first row.
Then,
finding algorithms and their computations are presented in
the survey paper [8].
and
(
includes zeros on
2) Obtaining
unit circle): Requires
CA and
CM.
3) Back substitutions (1) For
CM,
CA and
CD.
(2) For
CM,
CA and
CD. The low bandwidths of
and
are used here.
flops.
Hence the approximate complexity is
Depending on the computations for back-substitutions, order of
equalization can be interchanged to get further savings.
V. CHOICE OF THE ALGORITHM
A. Metric 1: Computational Complexity
Since
(order:
),
is invertible. Hence (5)
by back substitution, thereby recovcan be easily solved for
ering the transmitted symbols. Writing the entries of
in (5) in reverse order, we see that the time reversed input is
pumped on the time reversed channel
to obtain time reversed output (time reversed filtering). Note that
is minimum phase. As a result, in
the filter
the noisy case, the otherwise excess noise amplification due to
is tackled. A simtruncated causal filtering by
ilar idea is used in [9] for direct blind equalization. The order
of equalization can be interchanged and equalizers for the new
order can be similarly obtained. Unless specified otherwise, the
order is as detailed above.
Table I lists the approximate complexities of the three
schemes. We now try to find values of (given a ) for which
a given algorithm is cheaper. Finding such
by comparing
actual complexities listed in Sections III and IV is cumbersome. Instead, an analysis based on Table I is used to find the
for which an algorithm is cheaper.
approximate values of
Simulations are carried out to obtain numerical values of the
flops required by each of the three schemes. These simulations
aid the analysis in refining the earlier obtained values of .
We use “flops” command of Matlab Version 5.3 for counting
operations in the simulations. Adjustments are made to the
flop counts to account for real-complex operations. Since computations for min-max scheme depend on the actual channel
), average flop count obtained over 1000
(via
realizations of a random channel is presented. We present the
.
flop count ratios (for clarity) in Table II for a few
C. Zeros on Unit Circle
B. Metric 2: BER Performance
Adding zeros on the unit circle (say
of them) to
or/and
will not change the nonzero property of the
and
. Hence the earlier process can be used
resulting
and/or
. Thus Min-max equalization
with modified
possesses the ZFEQN property for all channels. However, in the
increases, performance of this
noisy case, it is seen that as
scheme departs from that of SZFE. When zeros are present on
the unit circle, SZFE can be used instead. However, for low ,
), gains obtained by using SZFE are small.
(say
We club the
zeros on unit circle to
in all further
discussions.
Since BBH and Householder’s methods are equivalent implementations of the SZFE, their BER curves coincide (the numerical properties of the two algorithms are assumed to be similar).
Hence, we present the BER curve of BBH method and refer it
to as the BER curve of SZFE. BER performance of Min-max
equalization is different from that of SZFE and needs to be presented. The BER curves are plotted in Figs. 1(a) and (b) for the
HiperLAN/2 channel models B and E respectively. QPSK modare used. BER curves are based on Monte
ulation and
Carlo simulations with each trial corresponding to a different
realization of the channel used. The channel is fixed for a frame
zeros) and varies across frames.
(
The channel is assumed to be estimated perfectly by the receiver
(till 5–8 dB), the perin every frame. At low values of
formance of Min-max equalizer is similar to that of SZFE. For
in the range 8–15 dB, Min-max equalizer requires about
1–1.5 dB of extra signal energy to have a similar BER as SZFE.
dB. We now
This offset rises to about 2–3 dB for
evaluate the “attractiveness” of the schemes described.
1) BBH Algorithm: For a given , Table I shows that the approximate complexity of BBH increases linearly (nearly) with
, while the increase is quadratic or cubic for the
(5)
D. Computational Complexity of Min-Max Equalizer
1) Finding the channel zeros: The roots of
are computed as the eigen-values of the companion matrix formed
[3] (roots command of Matlab). We approxiby
CA and
CM2. LAmate the complexity as
PACK manual notes the complexity as
. Better root2This involves Balancing, Hessenberg reduction followed by QR iterations
to obtain the Schur form. Each iteration results in O(L ) operations and the
maximum number of iterations is a function of L.
IEEE SIGNAL PROCESSING LETTERS, VOL. 11, NO. 9, SEPTEMBER 2004
755
TABLE I
SUMMARY OF APPROXIMATE FLOPS (A.F) OF VARIOUS SCHEMES
TABLE II
FLOP COUNT RATIOS OF VARIOUS SCHEMES FOR DIFFERENT (P; L). R =
Flop Count of Householder’s Method/Average Flip Count of Min–Max
Equalization. R = Flop Count of BBH Algorithm/Average Flop Count of
Min–Max Equalization
other schemes. Hence BBH algorithm is suited for large (i.e.,
such that
, for some
). Low value of
in
Table II for (64, 13), (128, 20), and (512, 50) shows that can
,
be chosen as 0.5.3 Since OFDM systems prefer a low
scope of this algorithm being attractive is limited.
2) Householder’s Method: From Table I we infer that
Householder’s method would be cheaper than BBH method for
for some
. Comparing the values of
and
in Table II and other simulations, we choose
. For
increases with as rooting is cheap in
small values of
the min-max scheme. For larger , computations for rooting
decreases with increase in . Furbecomes dominant and
ther, Table II and other simulations show that Householder’s
method hardly beats Min-max scheme in computations for
. However, the BER advantage [Figs. 1(a) and (b)]
, should be weighed
of this method, especially at high
against computational simplicity of Min-max equalization. This
results in a BER-Computation tradeoff, wherein the algorithm
needs to chosen based on system considerations.
3) Min-Max Equalization: For very small , rooting is
accurate and cheap [8]. For such , this scheme requires about
flops, which is less than the flops required by other al. Further Tables I, II, Section V.C.1
gorithms
and simulations show that min-max method is cheaper for
slightly less than
. However, BER performance
should also be evaluated to favor this algorithm. Figs. 1(a) and
, the BER performance of this
(b) show that for low
scheme is similar to that of SZFE. Thus for low and low
, Min-max equalization is attractive.
VI. CONCLUSION
In this work, we have used QR Decomposition to obtain low
complexity implementations of SZFE for the ZP scheme. In
particular, BBH and Householder’s algorithms are used. Motivated by equalization using FIR filters, we have developed
the Min-Max equalization scheme which yields a channel independent ZFE. The computational complexity and BER performance of each of these schemes are evaluated. While BBH
is the best for large , Min-max equalization is best for low
3While the Nagy’s algorithm [4], [5] is cheaper (by about (P =2) CA/CM)
y (i) than BBH, its flop count can be shown to be
in computing of
O(P ) + O(PL). Using Nagy’s algorithm instead of BBH, will only cause a
small change in without altering the spirit of the discussions.
Q
Fig. 1. BER Curves for HiperLAN/2 Channel models (a) B. (b) E.
and low
. For large
and small , we have a
BER-complexity trade-off between Householder’s method and
the Min-max scheme. The performance of these methods for imperfect channel estimates using their numerical properties [3],
[7], [8] needs a detailed study.
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